Order Cone Programming Formulation Research Articles

Finding an energy efficient path for plug-in electric vehicles with speed optimization and travel time restrictions

Transportation is one of the main factors when global total energy consumption is considered and is a significant contributor to emissions of harmful gases including carbon dioxide (CO2). Due to their lower tailpipe CO2 emissions compared to the vehicles with internal combustion engines, electric vehicles provide an opportunity to reduce environmental impacts of transportation. In this direction, a problem for plug-in electric vehicles (PEVs) is studied where the aim is to find an energy efficient path. Given an origin–destination pair over a directed network, this problem involves determining a path joining origin and destination, the speed of the PEV on each road segment, i.e., arc, along the path, the charging stations the PEV will stop by, and how much to recharge at each stop so as to minimize the total amount energy consumption. There are speed limits on each road segment, and PEV has to arrive at the destination on or before a given total time limit. For this problem, firstly, a mixed-integer second order cone programming formulation (MISOCP) is proposed. Secondly, to be able to solve larger size instances, a matheuristic is developed. Lastly, an iterated local search (ILS) algorithm is designed for this problem. Solution quality and computation times of the heuristics and the exact algorithm are compared on different instances. Differently from the literature, the speed values of the PEV on the arcs are considered as continuous decision variables in all proposed solution approaches. Moreover, consideration of the speed limits which can be legal limits or limits imposed by congestion makes our problem more realistic. The analysis of the results of the computational experiments gives the user an insight to select the proper solution approach based on the instance settings. MISOCP formulation becomes inadequate for larger instances. On the other hand, the heuristic solution approaches can solve such instances within reasonable computational times and therefore they have the potential to be integrated in some software to dynamically find energy efficient paths.

Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game

<p style='text-indent:20px;'>A vital issue in many organizations is the fair allocation of fixed cost among its subsets. In this paper, using data envelopment analysis, first we study fixed cost allocation based on both additive and multiplicative efficiency decompositions in the cooperative context for a two-stage structure in the presence of exogenous inputs and outputs. A conic relaxation formulation of multiplicative decomposition is given. Then, fixed cost allocation based on the leader-follower paradigm are presented. In the sequel, for allocating a fair fixed cost between the stages, using the results of the leader-follower model, we present the nonlinear Nash bargaining game model that independent of the efficiency score of each unit, allocates fixed cost to the stages. The nonlinear model is reformulated as a second order cone program which is an imporvement over the parametric linear models in the literature. Finally, two examples are used to illustrate the proposed models and compare their results with the existing models.</p>

Bicriteria scheduling of a material handling robot in an m-machine cell to minimize the energy consumption of the robot and the cycle time

This study considers a flowshop type production system consisting of m machines. A material handling robot transports the parts between the machines and loads and unloads the machines. We consider the sequencing of the robot moves and determining the speeds of these moves simultaneously. These decisions affect both the robot’s energy consumption and the production speed of the system. In this study, these two objectives are considered simultaneously. We propose a second order cone programming formulation to find Pareto efficient solutions. We also develop a heuristic algorithm that finds a set of approximate Pareto efficient solutions. The conic formulation can find robot schedules for small cells with less number of machines in reasonable computation times. Our heuristic algorithm can generate a large set of approximate Pareto efficient solutions in a very short computational time. Proposed solution approaches help the decision-maker to achieve the best trade-off between the throughput of a cell and the energy efficiency of a material handling robot.

Exact solution approaches for the workload smoothing in assembly lines

In this paper, the problem of minimizing the smoothness index for an assembly line given a fixed cycle time and the number of workstations is studied. This problem which is known as the workload smoothing line balancing problem (WSLBP) is a mixed-integer quadratic programming problem. Until recently, this problem has only been tackled using heuristic approaches. Recently, there have been some attempts to solve this problem exactly using mixed-integer linear programming (MILP). The MILP formulations, however, are not usually capable of solving large size problem instances. In this paper, the aim is to solve the WSLBP using mathematical programming formulations by using off-the-shelf solvers. Differently from the literature some non-MILP formulations are also considered for the problem. For this purpose, three MILP formulations, one from the literature, and two non-MILP formulations are compared. The two non-MILP formulations include a mixed-integer second order cone programming formulation and a constraint programming model. The superiority of the non-MILP formulations over the considered MILP formulations is experimentally shown.

A mixed integer programming formulation for the stochastic lot sizing problem with controllable processing times

In this study, we address the capacitated stochastic lot-sizing problem under α service level constraints. We assume that processing times can be decreased in return for compression cost that follows a convex function. We consider this problem under the static uncertainty strategy suggesting to determine replenishment plans at the beginning of the planning horizon. We develop an extended mixed integer programming (MIP) formulation built on a predefined piecewise linear approximation. Then, we adopt the so-called dynamic cut generation approach to be able to use the proposed MIP formulation with no prior approximation of the cost function. Also, we demonstrate how to extend the dynamic cut generation approach to consider the exact inventory cost in the objective function. We show the computational performance of the proposed MIP model with the dynamic cut generation approach in an extensive numerical study where second order cone programming formulations developed in the literature are used as benchmark. The results reveal that the proposed MIP model deployed with the dynamic cut generation yields a superior computational performance as compared to the benchmark formulations especially when the order of compression cost function is higher.

Green supply chain design with emission sensitive demand: second order cone programming formulation and case study

We propose a novel modeling framework for supply chain network design that models a prevailing trend in consumer choice in which demand is impacted by carbon footprint. To date, the literature lacks models that realistically account for and accurately calculate per unit emissions, i.e., carbon footprint. We develop a profit maximizing model that accounts for emissions at the different stages of the supply chain, locates facilities and selects their technology, and decides on the flow between echelons. To calculate the carbon footprint, fixed emissions are averaged over throughput, which results in a nonlinear optimization problem with fractional terms. To solve it, we provide a mixed integer second order cone programming reformulation. We perform extensive testing of the framework on a realistic case study and carry out detailed analysis. The proposed framework succeeds in capturing the trade-off between lost demand due to a high carbon footprint and investing in environmentally-friendly technology. The framework serves as a tool to induce organizations to invest in green technology and to allow regulating authorities to assess the impact of eco-labeling.

Robust CCMV model with short selling and risk-neutral interest rate

Robust optimization is a powerful tool for managing uncertainty in financial optimization problems. In this paper, we study the cardinality constraints mean–variance portfolio optimization model with short selling and risk-neutral interest rate under the return and covariance matrix uncertainty. Equivalent mixed integer second order cone programming formulation of its robust counterpart is given for the interval uncertainty. Finally, using S&P 500 index data, original and robust models are compared in terms of Sharpe ratios.

Solution methods for a min–max facility location problem with regional customers considering closest Euclidean distances

We study a facility location problem where a single facility serves multiple customers each represented by a (possibly non-convex) region in the plane. The aim of the problem is to locate a single facility in the plane so that the maximum of the closest Euclidean distances between the facility and the customer regions is minimized. Assuming that each customer region is mixed-integer second order cone representable, we firstly give a mixed-integer second order cone programming formulation of the problem. Secondly, we consider a solution method based on the Minkowski sums of sets. Both of these solution methods are extended to the constrained case in which the facility is to be located on a (possibly non-convex) subset of the plane. Finally, these two methods are compared in terms of solution quality and time with extensive computational experiments.

Profit-based credit scoring based on robust optimization and feature selection

A novel framework for profit-based credit scoring is proposed in this work. The approach is based on robust optimization, which is designed for dealing with uncertainty in the data, and therefore is effective at classifying new samples that follow a slightly different distribution in relation to the original dataset used to create the model. Instead of minimizing a loss function based on statistical measures, the proposed method maximizes the profit of the credit scoring model, balancing the benefits and losses of granting credit with the variable acquisition costs. The reduction of these is performed using feature selection techniques embedded in the learning process. The robust approach results in four second order cone programming formulations, which can be solved efficiently using interior point algorithms. Experiments on two credit scoring datasets demonstrate the virtues of our approach in terms of its predictive performance, and the managerial insights that can be gained from it.

Green hub location problem

This paper introduces the green hub location problem that finds the best locations for hubs, assignments of demand nodes to these hubs and speed of trucks/flights so as to route the demand between any origin-destination pairs. The aim of the service provider is to minimize the total amount of emissions that depends on vehicle speed and payload while routing the deliveries within a predetermined service time limit. In this study, we first propose a nonlinear model for this problem, which is then reformulated as a second order cone programming formulation. We strengthen the new model by using perspective reformulation approach. An extensive computational study on the CAB and TR datasets demonstrates the benefits of incorporating green transportation service activities to the classic hub location problems. We also provide insights for the carrier companies by analyzing the solutions with different discount factors, service time limits and number of hubs.

Linear Precoding to Optimize Throughput, Power Consumption and Energy Efficiency in MIMO Wireless Sensor Networks

This paper considers joint precoder design to optimize throughput, power consumption and energy efficiency (EE) in the context of multi-antenna wireless sensor networks with coherent multiple access channels. To maximize throughput, both centralized and decentralized algorithms are developed. Our centralized algorithm obtains a new second order cone programming formulation of the problem, which is different from related works and can apply to more generic system setup compared to existing literature. In addition, noting the fact that all existing solutions in literature are centralized based, we propose a novel decentralized solution and analyses its convergence. Besides the throughput maximization, the power consumption and EE problems are also attacked. To optimize these two metrics, a decentralized algorithm based on dual-decomposition and block successive upper-bound method has been developed, which runs in parallel with semi-analytical solutions and has provable strong convergence. A sufficient condition for the validity of the decentralized method is obtained. Extensive numerical results are presented to consolidate our findings.

Uncertainty-safe large scale support vector machines

The issue of large scale binary classification when data is subject to random perturbations is addressed. The proposed model integrates a learning framework that adjusts its robustness to noise during learning. The method avoids over-conservative situations that can be encountered with worst-case robust support vector machine formulations. The algorithm could be seen as a technique to learn the support-set of the noise distribution during the training process. This is achieved by introducing optimization variables that control the magnitude of the noise perturbations that should be taken into account. The magnitude is tuned by optimizing a generalization error. Only rough estimates of perturbations bounds are required. Additionally, a stochastic bi-level optimization technique is proposed to solve the resulting formulation. The algorithm performs very cheap stochastic subgradient moves and is therefore well suited to large datasets. Encouraging experimental results show that the technique outperforms robust second order cone programming formulations.

Continuous location under the effect of ‘refraction’

In this paper we address the problem of locating a new facility on a d-dimensional space when the distance measure ($$\ell _p$$lp- or polyhedral-norms) is different at each one of the sides of a given hyperplane $$\mathcal {H}$$H. We relate this problem with the physical phenomenon of refraction, and extend it to any finite dimensional space and different distances at each one of the sides of any hyperplane. An application to this problem is the location of a facility within or outside an urban area where different distance measures must be used. We provide a new second order cone programming formulation, based on the $$\ell _p$$lp-norm representation given in Blanco et al. (Comput Optim Appl 58(3):563---595, 2014) that allows to solve the problem in any finite dimensional space with second order cone or semidefinite programming tools. We also extend the problem to the case where the hyperplane is considered as a rapid transit media (a different third norm is also considered over $$\mathcal {H}$$H) that allows the demand to travel, whenever it is convenient, through $$\mathcal {H}$$H to reach the new facility. Extensive computational experiments run in Gurobi are reported in order to show the effectiveness of the approach. Some extensions of these models are also presented.

Robust twin support vector machine for pattern classification

In this paper, we proposed a new robust twin support vector machine (called R-TWSVM) via second order cone programming formulations for classification, which can deal with data with measurement noise efficiently. Preliminary experiments confirm the robustness of the proposed method and its superiority to the traditional robust SVM in both computation time and classification accuracy. Remarkably, since there are only inner products about inputs in our dual problems, this makes us apply kernel trick directly for nonlinear cases. Simultaneously we does not need to solve the extra inverse of matrices, which is totally different with existing TWSVMs. In addition, we also show that the TWSVMs are the special case of our robust model and simultaneously give a new dual form of TWSVM by degenerating R-TWSVM, which successfully overcomes the existing shortcomings of TWSVM.

Second Order Cone Programming Formulations for Handling Data with Perturbation

Ordinal regression problem and general multi-class classification problem are important and on-going research subject in machine learning. Support vector ordinal regression machine (SVORM) is an effective method for ordinal regression problem and has been used to deal with general multi-class classification problem. Up to now it is always assumed implicitly that the training data are known exactly . However, in practice, the training data subject to measurement noise. In this paper, we propose the robust versions of SVORM. Furthermore, we also propose a robust multi-class algorithm based on 3-class robust SVORM with Gaussian kernel for general multi-class classification problem with perturbation. The robustness of the proposed methods is validated by our preliminary numerical experiments.

Second Order Cone Programming Approaches for Handling Missing and Uncertain Data

We propose a novel second order cone programming formulation for designing robust classifiers which can handle uncertainty in observations. Similar formulations are also derived for designing regression functions which are robust to uncertainties in the regression setting. The proposed formulations are independent of the underlying distribution, requiring only the existence of second order moments. These formulations are then specialized to the case of missing values in observations for both classification and regression problems. Experiments show that the proposed formulations outperform imputation.

Second Order Cone Programming Formulations for Feature Selection

This paper addresses the issue of feature selection for linear classifiers given the moments of the class conditional densities. The problem is posed as finding a minimal set of features such that ...